While the first issue is a routine variant of [1]}, the second is not. In the nontrapping case, Graf [2]} and Hislop, Nakamura [3]} showed that
the classical nontrapping condition \(\lbrace a,h\rbrace >\alpha >0\)  on the entire energy level \(\lbrace h=E_0>0\rbrace \)  implies the Mourre estimate (REF ) for \(I\)  some small
interval around \(E_0\)  (in the semi-classical case with \(\hbar \)  sufficiently small).
In the case of surfaces of revolution as in Theorem REF  this fact, together with Theorem REF , implies that one can handle energies in the range \(\epsilon <E<V_{\rm max}(0)-\epsilon \)  since they verify a classical nontrapping condition. On the other hand, for energies near \(V_{\rm max}(0)\)  this fails since the top energy is
classically trapping. Nevertheless, the Heisenberg uncertainty principle (or the semiclassical harmonic oscillator) guarantee (REF ).
